The Effective One Body description of the Two-Body problem

TL;DR

The paper develops the Effective One Body (EOB) formalism to solve the two-body problem in general relativity and produce accurate gravitational-wave signals for binary black holes across all stages, via a resummed conservative dynamics encoded in $A(R)$ and a mapping between ${\mathcal{E}}_{\rm eff}$ and $E_{\rm real}$. It introduces a robust radiation-reaction and waveform resummation based on the factorized multipolar waveform with $\rho_{\ell m}$, validated by NR across mass ratios, achieving excellent phase and amplitude agreement. The results demonstrate that few free parameters, notably $(a_5,a_6)$ in the radial potential, suffice to match NR data up to merger and ringdown, and outline extensions to spins and tides for broader astrophysical relevance. This framework enables efficient generation of accurate gravitational-wave templates for current detectors and future space-based observatories, with potential applications to spinning and tidally deformed binaries.

Abstract

The Effective One Body (EOB) formalism is an analytical approach which aims at providing an accurate description of the motion and radiation of coalescing binary black holes with arbitrary mass ratio. We review the basic elements of this formalism and discuss its aptitude at providing accurate template waveforms to be used for gravitational wave data analysis purposes.

Figures (11)

• Figure 1: Sketch of the correspondence between the quantized energy levels of the real and effective conservative dynamics. $n$ denotes the 'principal quantum number' ($n = n_r + \ell + 1$, with $n_r = 0,1,\ldots$ denoting the number of nodes in the radial function), while $\ell$ denotes the (relative) orbital angular momentum $({\bm L}^2 = \ell (\ell + 1) \, \hbar^2)$. Though the EOB method is purely classical, it is conceptually useful to think in terms of the underlying (Bohr-Sommerfeld) quantization conditions of the action variables $I_R$ and $J$ to motivate the identification between $n$ and $\ell$ in the two dynamics.
• Figure 2: Various approximations and Padé resummation of the EOB radial potential $A(u)$, where $u=GM/(c^2R)$, for the equal-mass case $\nu=1/4$. The vertical dashed lines indicate the corresponding (adiabatic) LSO location Buonanno:1998gg defined by the condition $d^2{\cal E}^0_{\rm eff}/dR^2=d{\cal E}_{\rm eff}^0/dR=0$, where ${\cal E}^0_{\rm eff}$ is the effective energy along the sequence of circular orbits ( i.e., when $P_{R}^{\rm eff}=0$).
• Figure 3: The extreme-mass-ratio limit ($\nu=0$): the Newton-normalized energy flux emitted by a particle on circular orbits. The figure illustrates the scattering of the standard Taylor expansion of the flux around the "exact" numerical result (computed up to $\ell=6$) obtained via perturbation theory.
• Figure 4: The extreme-mass-ratio limit ($\nu=0$). Padé resummation of the Taylor expandend energy flux of Fig. \ref{['taylor']} as proposed in Ref. Damour:1997ub with $v_{\rm pole}=1/\sqrt{3}$. The sequence of Padé approximants is less scattered than the corresponding Taylor ones and closer to the exact result.
• Figure 5: The extreme mass ratio limit ($\nu=0$). Same of Fig. \ref{['pade_untuned']} but flexing the value of the parameter $v_{\rm pole}$ so to improve the agreement with the exact result.